Question

factor completely.
$9n^2 - 54n - 63$

Explanation:

Step1: Factor out the GCF

First, find the greatest common factor (GCF) of the terms \(9n^2\), \(-54n\), and \(-63\). The GCF of 9, 54, and 63 is 9. So we factor out 9:
\(9n^2 - 54n - 63 = 9(n^2 - 6n - 7)\)

Step2: Factor the quadratic trinomial

Now, factor the quadratic trinomial \(n^2 - 6n - 7\). We need two numbers that multiply to -7 and add up to -6. The numbers are -7 and 1, because \((-7) \times 1 = -7\) and \((-7) + 1 = -6\). So we can factor it as:
\(n^2 - 6n - 7 = (n - 7)(n + 1)\)

Step3: Combine the factors

Putting it all together, the completely factored form is:
\(9(n - 7)(n + 1)\)

Answer:

\(9(n - 7)(n + 1)\)